Question

The sum of the series $1+2x+3x^2+4x^3+.....$ upto n terms is

• A. $\frac{1-(n+1)x^n+n{x}^{n+1}}{(1-x{)}^2}$
• B. $\frac{1-x^n}{(1-x}$
• C. $x^{n+1}$
• D. None of these

Answer 1

The correct option is A $\frac{1-(n+1)x^n+n{x}^{n+1}}{(1-x{)}^2}$

Let $S_n$ be the sum of the given series to n terms, then
$S_n=1+2x+3x^2+4x^3+....+n{x}^{x-1}....\left(i\right)$
$x{S}_n=x+2x^2+3x^2+....+n{x}^n....\left(ii\right)$

Subtracting (ii) from (i), we get
$(1-x)S_n=1+x+x^2+x^3+....$ to n terms $-n{x}^n$
$=(\frac{(1-x^n)}{(1-x)}-n{x}^n)$
$\Rightarrow S_n=\frac{(1-x^n)-n{x}^n(1-x)}{(1-x{)}^2}=\frac{1-(n+1)x^n+n{x}^{n+1}}{(1-x{)}^2}.$